The toral Tits process of Jordan algebras
The paper is a continuation of the authors' work on exceptional simple Jordan algebras. The results obtained in the authors' papers ''Jordan algebras of degree 3 and the Tits process'' and ''Classification of algebras arising from the Tits process'', J. Algebra (to appear) show that 9-dimensional subalgebras of exceptional simple Jordan algebras are of special interest. In the authors' work such algebras have the form \(E\oplus (E\otimes C)\) for two separable commutative and associative algebras E,C of dimension 2,3 respectively, and the multiplication is described using the general Tits process. On the other hand, by structure theory such algebras are either of the form \(A^+\) or H(B,*) where A,B are simple associative algebras of degree 3 and * is an involution of the second kind. It is therefore a natural objective to derive a criterion for deciding when an algebra \(E\oplus (E\otimes C)\) is isomorphic to an \(A^+\) or H(B,*) respectively. The authors show: \(E\oplus (E\otimes C)\cong A^+\) if and only if \(\Delta (E)=\delta (C)\) where \(\delta\) (E) resp. \(\delta\) (C) are the discriminants which the authors associated to E and C in [Manuscr. Math. 45, 249-272 (1984; Zbl 0536.17009)]. The authors' criterion follows from a description of the center of the special universal envelope of \(E\oplus (E\otimes C)\).
scientific article; zbMATH DE number 3879115